arXiv 1102.47612011-02-23math.CO

有限个非负和实数集的极小问题

A Minimum problem for finite sets of real numbers with non-negative sum

Giampiero Chiaselotti, Giuseppe Marino, Caterina Nardi

备注：: 15 pages

摘要：

设$n$和$r$是两个整数，使得$0 < r \le n$ ;我们用$γ(n,r)$ [ $η(n,r)$ ]表示和$\sum_{1=1}^n a_i \ge 0$的非负部分和的最小[最大]数，其中$a_1, \cdots, a_n$是任意选择的$n$实数，以使它们中的$r$是非负的，其余的$n-r$是负的。受Manickam，Miklös和Singhi在1987年\cite{ManMik87}和1988年\cite{ManSin88}中提出的一些有趣的极值组合和问题的启发，我们研究了以下两个问题：\noindent $(P1)$ { \it是每个$n$和$r$的$γ(n,r)$和$η(n,r)$的值，$0 < r \le n$？} \noindent { \it如果$q$是一个整数，使得$γ(n,r) \le q \le η(n,r)$，我们能找到$n$实数$a_1, \cdots, a_n$，使得它们中的$r$是非负的，而其余的$n-r$是负的，与$\sum_{1=1}^n a_i \ge 0$，使得由这些数形成的非负和的个数正好是$q$？}证明了问题\noindent的解是由$γ(n,r) = 2^{n-1}$和$η(n,r) = 2^n - 2^{n-r}$给出的.我们提供了后一个问题的部分结果表明，答案是肯定的加权布尔映射。关于问题$(P2)$这样的地图（我们将在本文中介绍）可以被认为是一个推广的多集$a_1, \cdots, a_n$与$\sum_{1=1}^n a_i \ge 0$。更精确地说，我们证明了，对于每个$q$，使$γ(n,r) \le q \le η(n,r)$存在一个加权布尔映射，正好$q$正布尔值。

英文摘要：

Let $n$ and $r$ be two integers such that $0 < r \le n$; we denote by $γ(n,r)$ [$η(n,r)$] the minimum [maximum] number of the non-negative partial sums of a sum $\sum_{1=1}^n a_i \ge 0$, where $a_1, \cdots, a_n$ are $n$ real numbers arbitrarily chosen in such a way that $r$ of them are non-negative and the remaining $n-r$ are negative. Inspired by some interesting extremal combinatorial sum problems raised by Manickam, Miklös and Singhi in 1987 \cite{ManMik87} and 1988 \cite{ManSin88} we study the following two problems: \noindent$(P1)$ {\it which are the values of $γ(n,r)$ and $η(n,r)$ for each $n$ and $r$, $0 < r \le n$?} \noindent$(P2)$ {\it if $q$ is an integer such that $γ(n,r) \le q \le η(n,r)$, can we find $n$ real numbers $a_1, \cdots, a_n$, such that $r$ of them are non-negative and the remaining $n-r$ are negative with $\sum_{1=1}^n a_i \ge 0$, such that the number of the non-negative sums formed from these numbers is exactly $q$?} \noindent We prove that the solution of the problem $(P1)$ is given by $γ(n,r) = 2^{n-1}$ and $η(n,r) = 2^n - 2^{n-r}$. We provide a partial result of the latter problem showing that the answer is affirmative for the weighted boolean maps. With respect to the problem $(P2)$ such maps (that we will introduce in the present paper) can be considered a generalization of the multisets $a_1, \cdots, a_n$ with $\sum_{1=1}^n a_i \ge 0$. More precisely we prove that for each $q$ such that $γ(n,r) \le q \le η(n,r)$ there exists a weighted boolean map having exactly $q$ positive boolean values.

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